Contents

“Venus de Milo”, a famous ancient Greek sculpture. Note the complex lighting environment.

This tutorial introduces image-based lighting, in particular diffuse (irradiance) environment mapping and its implementation with cube maps. (Unity's Light Probes presumably work in a similar way but with dynamically rendered cube maps.)

Consider the lighting of the sculpture in the image to the left. There is natural light coming through the windows. Some of this light bounces off the floor, walls and visitors before reaching the sculpture. Additionally, there are artificial light sources, and their light is also shining directly and indirectly onto the sculpture. How many directional lights and point lights would be needed to simulate this kind of complex lighting environment convincingly? At least more than a handful (probably more than a dozen) and therefore the performance of the lighting computations is challenging.

This problem is addressed by image-based lighting. For static lighting environments that are described by an environment map, e.g. a cube map, image-based lighting allows us to compute the lighting by an arbitrary number of light sources with a single texture lookup in a cube map (see Section “Reflecting Surfaces” for a description of cube maps). How does it work?

In this section we focus on diffuse lighting. Assume that every texel (i.e. pixel) of a cube map acts as a directional light source. (Remember that cube maps are usually assumed to be infinitely large such that only directions matter, but positions don't.) The resulting lighting for a given surface normal direction can be computed as described in Section “Diffuse Reflection”. It's basically the cosine between the surface normal vector N and the vector to the light source L:

Since the texels are the light sources, L is just the direction from the center of the cube to the center of the texel in the cube map. A small cube map with 32×32 texels per face has already 32×32×6 = 6144 texels. Adding the illumination by thousands of light sources is not going to work in real time. However, for a static cube map we can compute the diffuse illumination for all possible surface normal vectors N in advance and store them in a lookup table. When lighting a point on a surface with a specific surface normal vector, we can then just look up the diffuse illumination for the specific surface normal vector N in that precomputed lookup table.

Thus, for a specific surface normal vector N we add (i.e. integrate) the diffuse illumination by all texels of the cube map. We store the resulting diffuse illumination for this surface normal vector in a second cube map (the “diffuse irradiance environment map” or “diffuse environment map” for short). This second cube map will act as a lookup table, where each direction (i.e. surface normal vector) is mapped to a color (i.e. diffuse illumination by potentially thousands of light sources). The fragment shader is therefore really simple (this one could use the vertex shader from Section “Reflecting Surfaces”):

It is just a lookup of the precomputed diffuse illumination using the surface normal vector of the rasterized surface point. However, the precomputation of the diffuse environment map is somewhat more complicated as described in the next section.

This section presents some JavaScript code to illustrate the computation of cube maps for diffuse (irradiance) environment maps. In order to use it in Unity, choose Create > JavaScript in the Project Window. Then open the script in Unity's text editor, copy the JavaScript code into it, and attach the script to the game object that has a material with the shader presented below. When a new readable cube map of sufficiently small dimensions is specified for the shader property _OriginalCube (which is labeled Environment Map in the shader user interface), the script will update the shader property _Cube (i.e. Diffuse Environment Map in the user interface) with a corresponding diffuse environment map. Note that the cube map has to be "readable," i.e. you have to check Readable in the Inspector when creating the cube map. Also note that you should use small cube maps of face dimensions 32×32 or smaller because the computation time tends to be very long for larger cube maps. Thus, when creating a cube map in Unity, make sure to choose a sufficiently small size.

The script includes only a handful of functions: Awake() initializes the variables; Update() takes care of communicating with the user and the material (i.e. reading and writing shader properties); computeFilteredCubemap() does the actual work of computing the diffuse environment map; and getDirection() is a small utility function for computeFilteredCubemap() to compute the direction associated with each texel of a cube map. Note that computeFilteredCubemap() not only integrates the diffuse illumination but also avoids discontinuous seams between faces of the cube map by setting neighboring texels along the seams to the same averaged color.

JavaScript code: click to show/hide

@scriptExecuteInEditMode()privatevaroriginalCubemap:Cubemap;// a reference to the // environment map specified in the shader by the userprivatevarfilteredCubemap:Cubemap;// the diffuse irradiance // environment map computed by this scriptfunctionUpdate(){varoriginalTexture:Texture=GetComponent(Renderer).sharedMaterial.GetTexture("_OriginalCube");// get the user-specified environment mapif(originalTexture==null)// did the user specify "None" for the environment map?{if(originalCubemap!=null){originalCubemap=null;filteredCubemap=null;GetComponent(Renderer).sharedMaterial.SetTexture("_Cube",null);}return;}elseif(originalTexture==originalCubemap&&filteredCubemap!=null&&null==GetComponent(Renderer).sharedMaterial.GetTexture("_Cube")){GetComponent(Renderer).sharedMaterial.SetTexture("_Cube",filteredCubemap);// set the computed diffuse environment map in the shader}elseif(originalTexture!=originalCubemap||filteredCubemap!=GetComponent(Renderer).sharedMaterial.GetTexture("_Cube"))// has the user specified a cube map that is different of // what we had processed previously?{if(EditorUtility.DisplayDialog("Processing of Environment Map","Do you want to process the cube map of face size "+originalTexture.width+"x"+originalTexture.width+"? (This will take some time.)","OK","Cancel"))// does the user really want to process this cube map?{originalCubemap=originalTexture;if(filteredCubemap!=GetComponent(Renderer).sharedMaterial.GetTexture("_Cube")){if(null!=GetComponent(Renderer).sharedMaterial.GetTexture("_Cube")){DestroyImmediate(GetComponent(Renderer).sharedMaterial.GetTexture("_Cube"));// clean up}}if(null!=filteredCubemap){DestroyImmediate(filteredCubemap);// clean up}computeFilteredCubemap();// compute the diffuse environment map GetComponent(Renderer).sharedMaterial.SetTexture("_Cube",filteredCubemap);// set the computed diffuse environment map in the shader}else// no cancel the processing and reset everything{originalCubemap=null;filteredCubemap=null;GetComponent(Renderer).sharedMaterial.SetTexture("_OriginalCube",null);GetComponent(Renderer).sharedMaterial.SetTexture("_Cube",null);}}return;}functioncomputeFilteredCubemap()// This function computes a diffuse environment map in // "filteredCubemap" of the same dimensions as "originalCubemap"// by integrating -- for each texel of "filteredCubemap" -- // the diffuse illumination from all texels of "originalCubemap" // for the surface normal vector corresponding to the direction // of each texel of "filteredCubemap".{filteredCubemap=Cubemap(originalCubemap.width,originalCubemap.format,true);// create the diffuse environment cube mapvarfilteredSize:int=filteredCubemap.width;varoriginalSize:int=originalCubemap.width;// compute all texels of the diffuse environment // cube map by iterating over all of themfor(varfilteredFace:int=0;filteredFace<6;filteredFace++){for(varfilteredI:int=0;filteredI<filteredSize;filteredI++){for(varfilteredJ:int=0;filteredJ<filteredSize;filteredJ++){varfilteredDirection:Vector3=getDirection(filteredFace,filteredI,filteredJ,filteredSize).normalized;vartotalWeight:float=0.0;varoriginalDirection:Vector3;varoriginalFaceDirection:Vector3;varweight:float;varfilteredColor:Color=Color(0.0,0.0,0.0);// sum (i.e. integrate) the diffuse illumination // by all texels in the original environment mapfor(varoriginalFace:int=0;originalFace<6;originalFace++){originalFaceDirection=getDirection(originalFace,1,1,3).normalized;// the normal vector of the facefor(varoriginalI:int=0;originalI<originalSize;originalI++){for(varoriginalJ:int=0;originalJ<originalSize;originalJ++){originalDirection=getDirection(originalFace,originalI,originalJ,originalSize);// direction to the texel, i.e. light sourceweight=1.0/originalDirection.sqrMagnitude;// take smaller size of more distant texels // into accountoriginalDirection=originalDirection.normalized;weight=weight*Vector3.Dot(originalFaceDirection,originalDirection);// take tilt of texels // compared to face into accountweight=weight*Mathf.Max(0.0,Vector3.Dot(filteredDirection,originalDirection));// directional filter for diffuse illuminationtotalWeight=totalWeight+weight;// instead of analytically normalization, // we just normalize to the potentially // maximum illuminationfilteredColor=filteredColor+weight*originalCubemap.GetPixel(originalFace,originalI,originalJ);// add the illumination by this texel}}}filteredCubemap.SetPixel(filteredFace,filteredI,filteredJ,filteredColor/totalWeight);// store the diffuse illumination of this texel}}}// Avoid seams between cube faces: // average edge texels to the same color on both sides of the seam // (except corner texels, see below)varmaxI:int=filteredCubemap.width-1;varaverage:Color;for(vari:int=1;i<maxI;i++){average=(filteredCubemap.GetPixel(0,i,0)+filteredCubemap.GetPixel(2,maxI,maxI-i))/2.0;filteredCubemap.SetPixel(0,i,0,average);filteredCubemap.SetPixel(2,maxI,maxI-i,average);average=(filteredCubemap.GetPixel(0,0,i)+filteredCubemap.GetPixel(4,maxI,i))/2.0;filteredCubemap.SetPixel(0,0,i,average);filteredCubemap.SetPixel(4,maxI,i,average);average=(filteredCubemap.GetPixel(0,i,maxI)+filteredCubemap.GetPixel(3,maxI,i))/2.0;filteredCubemap.SetPixel(0,i,maxI,average);filteredCubemap.SetPixel(3,maxI,i,average);average=(filteredCubemap.GetPixel(0,maxI,i)+filteredCubemap.GetPixel(5,0,i))/2.0;filteredCubemap.SetPixel(0,maxI,i,average);filteredCubemap.SetPixel(5,0,i,average);average=(filteredCubemap.GetPixel(1,i,0)+filteredCubemap.GetPixel(2,0,i))/2.0;filteredCubemap.SetPixel(1,i,0,average);filteredCubemap.SetPixel(2,0,i,average);average=(filteredCubemap.GetPixel(1,0,i)+filteredCubemap.GetPixel(5,maxI,i))/2.0;filteredCubemap.SetPixel(1,0,i,average);filteredCubemap.SetPixel(5,maxI,i,average);average=(filteredCubemap.GetPixel(1,i,maxI)+filteredCubemap.GetPixel(3,0,maxI-i))/2.0;filteredCubemap.SetPixel(1,i,maxI,average);filteredCubemap.SetPixel(3,0,maxI-i,average);average=(filteredCubemap.GetPixel(1,maxI,i)+filteredCubemap.GetPixel(4,0,i))/2.0;filteredCubemap.SetPixel(1,maxI,i,average);filteredCubemap.SetPixel(4,0,i,average);average=(filteredCubemap.GetPixel(2,i,0)+filteredCubemap.GetPixel(5,maxI-i,0))/2.0;filteredCubemap.SetPixel(2,i,0,average);filteredCubemap.SetPixel(5,maxI-i,0,average);average=(filteredCubemap.GetPixel(2,i,maxI)+filteredCubemap.GetPixel(4,i,0))/2.0;filteredCubemap.SetPixel(2,i,maxI,average);filteredCubemap.SetPixel(4,i,0,average);average=(filteredCubemap.GetPixel(3,i,0)+filteredCubemap.GetPixel(4,i,maxI))/2.0;filteredCubemap.SetPixel(3,i,0,average);filteredCubemap.SetPixel(4,i,maxI,average);average=(filteredCubemap.GetPixel(3,i,maxI)+filteredCubemap.GetPixel(5,maxI-i,maxI))/2.0;filteredCubemap.SetPixel(3,i,maxI,average);filteredCubemap.SetPixel(5,maxI-i,maxI,average);}// Avoid seams between cube faces: average corner texels // to the same color on all three faces meeting in one corneraverage=(filteredCubemap.GetPixel(0,0,0)+filteredCubemap.GetPixel(2,maxI,maxI)+filteredCubemap.GetPixel(4,maxI,0))/3.0;filteredCubemap.SetPixel(0,0,0,average);filteredCubemap.SetPixel(2,maxI,maxI,average);filteredCubemap.SetPixel(4,maxI,0,average);average=(filteredCubemap.GetPixel(0,maxI,0)+filteredCubemap.GetPixel(2,maxI,0)+filteredCubemap.GetPixel(5,0,0))/3.0;filteredCubemap.SetPixel(0,maxI,0,average);filteredCubemap.SetPixel(2,maxI,0,average);filteredCubemap.SetPixel(5,0,0,average);average=(filteredCubemap.GetPixel(0,0,maxI)+filteredCubemap.GetPixel(3,maxI,0)+filteredCubemap.GetPixel(4,maxI,maxI))/3.0;filteredCubemap.SetPixel(0,0,maxI,average);filteredCubemap.SetPixel(3,maxI,0,average);filteredCubemap.SetPixel(4,maxI,maxI,average);average=(filteredCubemap.GetPixel(0,maxI,maxI)+filteredCubemap.GetPixel(3,maxI,maxI)+filteredCubemap.GetPixel(5,0,maxI))/3.0;filteredCubemap.SetPixel(0,maxI,maxI,average);filteredCubemap.SetPixel(3,maxI,maxI,average);filteredCubemap.SetPixel(5,0,maxI,average);average=(filteredCubemap.GetPixel(1,0,0)+filteredCubemap.GetPixel(2,0,0)+filteredCubemap.GetPixel(5,maxI,0))/3.0;filteredCubemap.SetPixel(1,0,0,average);filteredCubemap.SetPixel(2,0,0,average);filteredCubemap.SetPixel(5,maxI,0,average);average=(filteredCubemap.GetPixel(1,maxI,0)+filteredCubemap.GetPixel(2,0,maxI)+filteredCubemap.GetPixel(4,0,0))/3.0;filteredCubemap.SetPixel(1,maxI,0,average);filteredCubemap.SetPixel(2,0,maxI,average);filteredCubemap.SetPixel(4,0,0,average);average=(filteredCubemap.GetPixel(1,0,maxI)+filteredCubemap.GetPixel(3,0,maxI)+filteredCubemap.GetPixel(5,maxI,maxI))/3.0;filteredCubemap.SetPixel(1,0,maxI,average);filteredCubemap.SetPixel(3,0,maxI,average);filteredCubemap.SetPixel(5,maxI,maxI,average);average=(filteredCubemap.GetPixel(1,maxI,maxI)+filteredCubemap.GetPixel(3,0,0)+filteredCubemap.GetPixel(4,0,maxI))/3.0;filteredCubemap.SetPixel(1,maxI,maxI,average);filteredCubemap.SetPixel(3,0,0,average);filteredCubemap.SetPixel(4,0,maxI,average);filteredCubemap.Apply();// apply all the texture.SetPixel(...) commands}functiongetDirection(face:int,i:int,j:int,size:int):Vector3// This function computes the direction that is // associated with a texel of a cube map {vardirection:Vector3;if(face==0){direction=Vector3(0.5,-((j+0.5)/size-0.5),-((i+0.5)/size-0.5));}elseif(face==1){direction=Vector3(-0.5,-((j+0.5)/size-0.5),((i+0.5)/size-0.5));}elseif(face==2){direction=Vector3(((i+0.5)/size-0.5),0.5,((j+0.5)/size-0.5));}elseif(face==3){direction=Vector3(((i+0.5)/size-0.5),-0.5,-((j+0.5)/size-0.5));}elseif(face==4){direction=Vector3(((i+0.5)/size-0.5),-((j+0.5)/size-0.5),0.5);}elseif(face==5){direction=Vector3(-((i+0.5)/size-0.5),-((j+0.5)/size-0.5),-0.5);}returndirection;}

As an alternative to the JavaScript code above, you can also use the following C# code. Make sure to call the script file "ComputeDiffuseEnvironmentMap".

C# code: click to show/hide

usingUnityEngine;usingUnityEditor;usingSystem.Collections;[ExecuteInEditMode]publicclassComputeDiffuseEnvironmentMap:MonoBehaviour{publicCubemaporiginalCubeMap;// environment map specified in the shader by the user//[System.Serializable] // avoid being deleted by the garbage collector, // and thus leakingprivateCubemapfilteredCubeMap;// the computed diffuse irradience environment map privatevoidUpdate(){CubemaporiginalTexture=null;try{originalTexture=GetComponent<Renderer>().sharedMaterial.GetTexture("_OriginalCube")asCubemap;}catch(System.Exception){Debug.LogError("'_OriginalCube' not found on shader. "+"Are you using the wrong shader?");return;}if(originalTexture==null)// did the user set "none" for the map?{if(originalCubeMap!=null){GetComponent<Renderer>().sharedMaterial.SetTexture("_Cube",null);originalCubeMap=null;filteredCubeMap=null;return;}}elseif(originalTexture==originalCubeMap&&filteredCubeMap!=null&&GetComponent<Renderer>().sharedMaterial.GetTexture("_Cube")==null){GetComponent<Renderer>().sharedMaterial.SetTexture("_Cube",filteredCubeMap);// set the computed // diffuse environment map in the shader}elseif(originalTexture!=originalCubeMap||filteredCubeMap!=GetComponent<Renderer>().sharedMaterial.GetTexture("_Cube")){if(EditorUtility.DisplayDialog("Processing of Environment Map","Do you want to process the cube map of face size "+originalTexture.width+"x"+originalTexture.width+"? (This will take some time.)","OK","Cancel")){if(filteredCubeMap!=GetComponent<Renderer>().sharedMaterial.GetTexture("_Cube")){if(GetComponent<Renderer>().sharedMaterial.GetTexture("_Cube")!=null){DestroyImmediate(GetComponent<Renderer>().sharedMaterial.GetTexture("_Cube"));// clean up}}if(filteredCubeMap!=null){DestroyImmediate(filteredCubeMap);// clean up}originalCubeMap=originalTexture;filteredCubeMap=computeFilteredCubeMap();//computes the diffuse environment mapGetComponent<Renderer>().sharedMaterial.SetTexture("_Cube",filteredCubeMap);// set the computed // diffuse environment map in the shaderreturn;}else{originalCubeMap=null;filteredCubeMap=null;GetComponent<Renderer>().sharedMaterial.SetTexture("_Cube",null);GetComponent<Renderer>().sharedMaterial.SetTexture("_OriginalCube",null);}}}// This function computes a diffuse environment map in // "filteredCubemap" of the same dimensions as "originalCubemap"// by integrating -- for each texel of "filteredCubemap" -- // the diffuse illumination from all texels of "originalCubemap" // for the surface normal vector corresponding to the direction // of each texel of "filteredCubemap".privateCubemapcomputeFilteredCubeMap(){CubemapfilteredCubeMap=newCubemap(originalCubeMap.width,originalCubeMap.format,true);intfilteredSize=filteredCubeMap.width;intoriginalSize=originalCubeMap.width;// Compute all texels of the diffuse environment cube map // by itterating over all of themfor(intfilteredFace=0;filteredFace<6;filteredFace++)// the six sides of the cube{for(intfilteredI=0;filteredI<filteredSize;filteredI++){for(intfilteredJ=0;filteredJ<filteredSize;filteredJ++){Vector3filteredDirection=getDirection(filteredFace,filteredI,filteredJ,filteredSize).normalized;floattotalWeight=0.0f;Vector3originalDirection;Vector3originalFaceDirection;floatweight;ColorfilteredColor=newColor(0.0f,0.0f,0.0f);// sum (i.e. integrate) the diffuse illumination // by all texels in the original environment mapfor(intoriginalFace=0;originalFace<6;originalFace++){originalFaceDirection=getDirection(originalFace,1,1,3).normalized;//the normal vector of the facefor(intoriginalI=0;originalI<originalSize;originalI++){for(intoriginalJ=0;originalJ<originalSize;originalJ++){originalDirection=getDirection(originalFace,originalI,originalJ,originalSize);// direction to the texel // (i.e. light source)weight=1.0f/originalDirection.sqrMagnitude;// take smaller size of more // distant texels into accountoriginalDirection=originalDirection.normalized;weight=weight*Vector3.Dot(originalFaceDirection,originalDirection);// take tilt of texel compared // to face into accountweight=weight*Mathf.Max(0.0f,Vector3.Dot(filteredDirection,originalDirection));// directional filter // for diffuse illuminationtotalWeight=totalWeight+weight;// instead of analytically // normalization, we just normalize // to the potential max illuminationfilteredColor=filteredColor+weight*originalCubeMap.GetPixel((CubemapFace)originalFace,originalI,originalJ);// add the // illumination by this texel }}}filteredCubeMap.SetPixel((CubemapFace)filteredFace,filteredI,filteredJ,filteredColor/totalWeight);// store the diffuse illumination of this texel}}}// Avoid seams between cube faces: average edge texels // to the same color on each side of the seamintmaxI=filteredCubeMap.width-1;for(inti=0;i<maxI;i++){setFaceAverage(reffilteredCubeMap,0,i,0,2,maxI,maxI-i);setFaceAverage(reffilteredCubeMap,0,0,i,4,maxI,i);setFaceAverage(reffilteredCubeMap,0,i,maxI,3,maxI,i);setFaceAverage(reffilteredCubeMap,0,maxI,i,5,0,i);setFaceAverage(reffilteredCubeMap,1,i,0,2,0,i);setFaceAverage(reffilteredCubeMap,1,0,i,5,maxI,i);setFaceAverage(reffilteredCubeMap,1,i,maxI,3,0,maxI-i);setFaceAverage(reffilteredCubeMap,1,maxI,i,4,0,i);setFaceAverage(reffilteredCubeMap,2,i,0,5,maxI-i,0);setFaceAverage(reffilteredCubeMap,2,i,maxI,4,i,0);setFaceAverage(reffilteredCubeMap,3,i,0,4,i,maxI);setFaceAverage(reffilteredCubeMap,3,i,maxI,5,maxI-i,maxI);}// Avoid seams between cube faces: // average corner texels to the same color // on all three faces meeting in one cornersetCornerAverage(reffilteredCubeMap,0,0,0,2,maxI,maxI,4,maxI,0);setCornerAverage(reffilteredCubeMap,0,maxI,0,2,maxI,0,5,0,0);setCornerAverage(reffilteredCubeMap,0,0,maxI,3,maxI,0,4,maxI,maxI);setCornerAverage(reffilteredCubeMap,0,maxI,maxI,3,maxI,maxI,5,0,maxI);setCornerAverage(reffilteredCubeMap,1,0,0,2,0,0,5,maxI,0);setCornerAverage(reffilteredCubeMap,1,maxI,0,2,0,maxI,4,0,0);setCornerAverage(reffilteredCubeMap,1,0,maxI,3,0,maxI,5,maxI,maxI);setCornerAverage(reffilteredCubeMap,1,maxI,maxI,3,0,0,4,0,maxI);filteredCubeMap.Apply();//apply all SetPixel(..) commandsreturnfilteredCubeMap;}privatevoidsetFaceAverage(refCubemapfilteredCubeMap,inta,intb,intc,intd,inte,intf){Coloraverage=(filteredCubeMap.GetPixel((CubemapFace)a,b,c)+filteredCubeMap.GetPixel((CubemapFace)d,e,f))/2.0f;filteredCubeMap.SetPixel((CubemapFace)a,b,c,average);filteredCubeMap.SetPixel((CubemapFace)d,e,f,average);}privatevoidsetCornerAverage(refCubemapfilteredCubeMap,inta,intb,intc,intd,inte,intf,intg,inth,inti){Coloraverage=(filteredCubeMap.GetPixel((CubemapFace)a,b,c)+filteredCubeMap.GetPixel((CubemapFace)d,e,f)+filteredCubeMap.GetPixel((CubemapFace)g,h,i))/3.0f;filteredCubeMap.SetPixel((CubemapFace)a,b,c,average);filteredCubeMap.SetPixel((CubemapFace)d,e,f,average);filteredCubeMap.SetPixel((CubemapFace)g,h,i,average);}privateVector3getDirection(intface,inti,intj,intsize){switch(face){case0:returnnewVector3(0.5f,-((j+0.5f)/size-0.5f),-((i+0.5f)/size-0.5f));case1:returnnewVector3(-0.5f,-((j+0.5f)/size-0.5f),((i+0.5f)/size-0.5f));case2:returnnewVector3(((i+0.5f)/size-0.5f),0.5f,((j+0.5f)/size-0.5f));case3:returnnewVector3(((i+0.5f)/size-0.5f),-0.5f,-((j+0.5f)/size-0.5f));case4:returnnewVector3(((i+0.5f)/size-0.5f),-((j+0.5f)/size-0.5f),0.5f);case5:returnnewVector3(-((i+0.5f)/size-0.5f),-((j+0.5f)/size-0.5f),-0.5f);default:returnVector3.zero;}}}

The shader and script above are sufficient to compute diffuse illumination by a large number of static, directional light sources. But what about the specular illumination discussed in Section “Specular Highlights”, i.e.:

With this equation, we can compute a lookup table (i.e. a cube map) that contains the specular illumination by many light sources for any reflected view vector Rview{\displaystyle _{\text{view}}}. In order to look up the specular illumination with such a table, we just need to compute the reflected view vector and perform a texture lookup in a cube map. In fact, this is exactly what the shader code of Section “Reflecting Surfaces” does. Thus, we actually only need to compute the lookup table.

It turns out that the JavaScript code presented above can be easily adapted to compute such a lookup table. All we have to do is to change the line

where 50.0 should be replaced by a variable for nshininess{\displaystyle n_{\text{shininess}}}. This allows us to compute lookup tables for any specific shininess. (The same cube map could be used for varying values of the shininess if the mipmap-level was specified explicitly using the textureCubeLod instruction in the shader; however, this technique is beyond the scope of this tutorial.)

about (dynamic) diffuse environment maps, you could read Chapter 10, “Real-Time Computation of Dynamic Irradiance Environment Maps” by Gary King of the book “GPU Gems 2” by Matt Pharr (editor) published 2005 by Addison-Wesley, which is available online.